OPTIMAL MULTIPLE STOPPING TIME PROBLEM

We study the optimal multiple stopping time problem defined for each stopping time S by v(S) = ess sup(tau 1), ... , (tau d) (>=) (S) E[psi(tau(1), ... , tau(d))vertical bar F-S]. The key point is the construction of a new reward phi such that the value function v(S) also satisfies v(S) = ess sup(theta >= S) E[phi(theta)vertical bar F-S]. This new reward phi is not a right-continuous adapted process as in the classical case, but a family of random variables. For such a reward, we prove a new existence result for optimal stopping times under weaker assumptions than in the classical case. This result is used to prove the existence of optimal multiple stopping times for v(S) by a constructive method. Moreover, under strong regularity assumptions on psi, we show that the new reward phi can be aggregated by a progressive process. This leads to new applications, particularly in finance (applications to American options with multiple exercise times).